Find all values $\displaystyle a, b, c$ given the equations below such that there (a) exists exactly 1 solution (b) exists an infinite # of solutions (c) exists no solution

$\displaystyle x_1 + 5x_2 + x_3 = 0$

$\displaystyle x_1 + 6x_2 - x_3 = 0$

$\displaystyle 2x_1 + ax_2 + bx_3 = c$

My work for this problem:

We have $\displaystyle \left[ \begin {array}{cccc} 1&5&1&0\\\noalign{\medskip}1&6&-1&0\\\noalign{\medskip}2&a&b&c\end {array} \right]$

Using Maple to get it in RREF we have:

$\displaystyle \left[ \begin {array}{cccc} 1&0&0&{\frac {-11c}{b-22+2\,a}}\\\noalign{\medskip}0&1&0&{\frac {2c}{b-22+2\,a}}\\\noalign{\medskip}0&0&1&{\frac {c}{b-22+2\,a}}\end {array} \right]$

It looks like it will always have a solution (since there is a pivot in every row). And the only way to get an infinite # of solutions would be to have a free variable. So I'm not sure what to do with this problem.